Partitions of a Finite Partially Ordered Set

Abstract

In this paper, we investigate the notion of partition of a finite partially ordered set (poset, for short). We will define three different notions of partition of a poset, namely, monotone, regular, and open partition. For each of these notions we will find three equivalent definitions, that will be shown to be equivalent. We start by defining partitions of a poset in terms of fibres of some surjection having the poset as domain. We then obtain combinatorial characterisations of such notions in terms of blocks, without reference to surjection. Finally, we give a further, equivalent definition of each kind of partition by means of analogues of equivalence relations.Comment: Preprint submitted for publication in the book "From Combinatorics to Philosophy: The Legacy of G.-C. Rota

Reconfigurable Decorated PT Nets with Inhibitor Arcs and Transition Priorities

“…We first need to investigate the category PoSetsof partially ordered sets. In [15] this category has been examined. Definition 8 (Category PoSets).…”

“…Composition and identity are defined as for sets and are both order-preserving, PoSetsis indeed a category [15].…”

“…Next we introduce the subclass of monomorphisms M. Monomorphisms in PoSets are the injective order preserving maps [15] and order embeddingsthose mappings that satisfy item 1 in Def. 9 -are regular monomorphisms [15]. Definition 9 (Class M).…”